Global color variations on the Martian surface

Surface materials exposed throughout the equatorial region of Mars have been classified and mapped on the basis of spectral reflectance properties determined by the Viking II Orbiter vidicon cameras. Frames acquired at each of three wavelengths (0.45 ± 0.03 μm, 0.53 ± 0.05 μm, and 0.59 ± 0.05 μm) during the approach of Viking Orbiter II in Martian summer (L_s = 105°) were mosaicked by computer. The mosaics cover latitudes 30°N to 63°S for 360° of longitude and have resolutions between 10 and 20 km per line pair. Image processing included Mercator transformation and removal of an average Martian photometric function to produce albedo maps at three wavelengths. The classical dark region between the equator and ～30°S in the Martian highlands is composed of two units: (i) and ancient unit consisting of topographic highs (ridges, crater rims, and rugged plateaus riddled with small dendritic channels) which is among the reddest on the planet $\text{(0.59/0.45 μ}\text{m}\text{? 3)}$; and (ii) intermediate age, smooth, intercrater volcanic plains displaying numerous mare ridges which are among the least red on Mars $\text{(0.59/0.45 μ}\text{m}\text{? 2)}$. The relatively young shield volcanoes are, like the oldest unit, dark and very red. Two probable eolian deposits are recognized in the intermediate and high albedo regions. The stratigraphically lower unit is intermediate in both color $\text{(0.59/ 0.45 μ}\text{m}\text{? 2.5)}$ and albedo. The upper unit has the highest albedo, is very red $\text{(0.59/0.45 μ}\text{m}\text{? 3)}$, and is apparently the major constituent of the annual dust storms as its areal extent changes from year to year. The south polar ice cap and condensate clouds dominate the southernmost part of the mosaics.     

The clouds of Venus: An anisotropic scattering model

The phase function $\text{ω}\text{?}\text{(1 + a}\text{cos}\text{θ)}$ for anisotropic scattering is applied to a homogeneous atmospheric model to ascertain the effects of anisotropy on the near-infrared spectrum of Venus. L. D. G. Young's equivalent widths for the 820 Å CO₂ band are analyzed to derive allowed combinations of CO₂ specific abundance, continuum albedo, pressure, and degree of anisotropy. From these combinations, values are derived for the photon mean-free path and total optical thickness of the clouds. The Venera 10 measurement of the transmitted flux at 7200 Å is then used in conjunction with spherical albedo requirements to reduce the range of possible solutions to the equivalent-width analysis. Through the application of approximate similarity relations, the “true” (i.e., anisotropic) atmospheric parameters are derived from the isotropic values. For an assumed Mie scatterer with an average anisotropy factor of 〈cos θ〉 = 0.7, the results indicate a total optical thickness of about 55 with a single-scattering albedo of 0.9992 to 0.9995.     

Zodiacal light gathered along the line of sight: Retrieval of the local scattering coefficient from photometric surveys of the ecliptic plane

A clue towards a retrieval of the zodiacal brightness gathering along a line of sight in the ecliptic plane consists in introducing the other intersection of that line with the terrestrial orbit (Fig. 1). The distribution of the elemental contribution to the brightness, or of the local quantity $\text{D}$ [directional scattering coefficient, i.e. cross-section of the unit-volume, which gives very simple expressions (1), (2) for the brightness integral] can then be approached with reduced uncertainty. The assumptions-steady state of the zodiacal cloud; smooth distribution of $\text{D}$—are strongly suggested by the observations, and are much less controversial than the classical assumption of uniform composition and size everywhere.The scattering coefficient may vary along the line of sight as seen in Fig. 3 : an uncertainty bar highly dependent of the abscissa, and considerably reduced in the vicinity of two “nodes”. Both in abscissae and in ordinates, these nodes are conspicuously insensitive to the arbitrary choice of a mathematical model (Table 1).The node exterior to the Earth's orbit (“martian node”) remains at r ? 1.5 a.u. from the Sun (Fig. 4). It gives access to a range of the scattering phase function near Mars' orbit, deconvolved from any radial dependence of that function (Fig. 5). The backscattering effect obtained is a new confirmation of the non-terrestrial origin of the gegenschein.The node interior to the Earth's orbit remains located not far from the middle of each chord (“quasi-radial node”. Fig. 4). It allows to retrieve the radial dependence of $\text{D}$, partly deconvolved from its angular dependence, between 0.5 and 1 a.u. (Fig. 6 and Table 4).The uncertainty bars on $\text{D}$ at the two observing locations yield two uncertainty bars of the phase function σ(θ) at 1 a.u. (Fig. 7). At θ = 30°, the forward scattering efficiency (normalized to θ = 90°) cannot exceed 6 and more likely 4. This disagrees with higher values obtained assuming spherical particles, and even obtained in part of the more realistic studies (assuming irregularly shaped particles, or mainly observational) reviewed in Table 5.All of these results are derived, with fair agreement, from three independent observational sources.     

A laboratory study of the bidirectional reflectance from particulate samples

The Hapke (Hapke, B. [1981]. J. Geophys. Res. 86, 3039-3054) photometric model and its modifications are widely used to characterize telescopic, spacecraft, and laboratory observations of the bidirectional reflectance of particulate surfaces. Following work and methods laid out in a companion paper (Helfenstein, P., Shepard, M.K. [2011]. Icarus, in press), we deconstruct the Hapke model and, separating all empirical and ad hoc parameters (opposition surge, particle phase function, surface roughness), combine them into a single parameter called the surface phase function, F(α). We illustrate how to extract this function from scattering data sets acquired with the Bloomsburg University Goniometer (BUG). We show how this method can be used to rapidly and accurately characterize bidirectional reflectance data sets from laboratory and spacecraft measurements, often giving better fits to the data. We examine samples with strong color contrasts in different wavelengths. This allows us to examine the exact same surface, changing only the albedo to investigate how the amplitude and the detailed shape of the surface phase function might systematically depend on wavelength and albedo. We also examine the changes in scattering behavior that result when samples are compacted and find the surface phase function and single scattering albedo to be significantly changed. We suggest that these observations support the hypothesis that much of the scattering behavior attributed to the single particle phase function is instead cause by the surface micro-structure.     

First measurements with the Physikalisches Institut Radiometric Experiment (PHIRE)

We have constructed an experiment to perform bidirectional reflectance distribution function (BRDF) measurements of laboratory samples, and have used the experiment to characterize a sample of JSC-1 lunar regolith simulant. Characterizations relied on in-plane BRDF measurements in visible and near-infrared (NIR) bandpasses. The optical properties of the simulant sample were found to be similar to those observed for bright, lunar highland regions. Reflectance models (Hapke 1981. Bidirectional reflectance spectroscopy 1. Theory. J. Geophys. Res. 86(B4), 3,039−3,054; 1984. Bidirectional reflectance spectroscopy 3. Correction for macroscopic roughness. Icarus 59, 41−59; 1986. Bidirectional reflectance spectroscopy 4. The extinction coefficient and the opposition effect. Icarus 67, 264−280; 2002. Bidirectional reflectance spectroscopy 5. The coherent backscatter opposition effect and anisotropic scattering. Icarus 157, 523−534) made excellent fits to fixed incidence angle, variable emission angle data sets. However, the models were not found to extrapolate well to fixed, near-zero phase angle data at varying incidence angles, and no solutions were found that provided simultaneous, high quality fits to the two types of data sets. Except for the single-scattering albedo, the best-fit parameters of the fixed incidence angle data were statistically the same in the visible and NIR. Correlations between the reflectance model parameters were systematically examined, and strong correlations were found between single-scattering albedo and the two two-stream Henyey-Greenstein scattering parameters and, to a lesser extent, the small-scale mean surface roughness.     

The range of validity of the Eddington approximation

The Eddington approximation is often assumed to be useful only for optically thick media having a single-scattering albedo near unity. We present detailed evidence in this paper that, for homogeneous layers illuminated by a beam of radiation, the Eddington approximation predicts albedo and absorptivity reasonably well for all values of optical depth and single-scattering albedo, for several scattering phase functions (Rayleigh, Henyey-Greenstein, and Mie) having asymmetry factors less than or equal to $\text{1}\text{2}$. The worst errors are in the neighborhood of optical depth unity and single-scattering albedo 0.5. The Eddington approximation is further found to maintain good accuracy over almost the full range of incident beam directions and surface albedos. It is least accurate for the Mie phase function example, where one can obtain a dramatic improvement in accuracy by going over to the δ-Eddington approximation; this shows that the forward peak of the Mie phase function, and not its detailed shape, is the primary cause of diminished accuracy in the Eddington approximation.     

Limb darkening of meteorites and asteroids

This paper presents the results of a laboratory study of the limb darkening, near opposition, of the carbonaceous chondrites Orgueil (C1), Murchison (C2), and Allende (C3), the ordinary chondrite Bruderheim (L6), and a stainless-steel powder. These materials represent possible analogs for the surface materials of C, S, and M asteroids respectively. At low phase angles, the limb-darkening behavior of all materials studied is well represented by Minnaert's law. For carbonaceous chondrites, the Minnaert limb-darkening parameter k is nearly independent of wavelength for wavelengths between 0.4 and 0.9 μm, with a typical value of k = 0.55. The reflectance parameter, B₀, varies from 0.045 to 0.065 over the same range of wavelengths. Both k and B₀ are larger for the stainless-steel powder and the ordinary chondrite, due to the increased importance of multiple scattering in the surface layer. If no limb darkening were present, k would equal $\text{1}\text{2}$ and the geometric albedo (p) of an asteroid would equal the normal reflectance $\text{(r}{}_{\mathrm{<mtext>n</mtext>}}\text{? B}{}_0\text{)}$ of its surface material. For bodies whose surface material is appreciably limb darkened, the geometric albedo measured at the telescope will be lower than the true normal reflectance of surface material; we estimate that for S and M objects r_n ? 1.05 p. In the case of nonspherical asteroids, because the distribution of incidence and emission angles varies as the asteroid rotates, the geometric albedo must change with aspect. If limb darkening is not considered when interpreting asteroid light curves, the values of b/a derived will be too extreme. This effect is probably too small to be observed for C asteroids, because of their intrinsically low reflectances, but could be appreciable for S and M objects.     

Infrared (0.83-5.1 μm) photometry of Phoebe from the Cassini Visual Infrared Mapping Spectrometer

Three weeks prior to the commencement of Cassini's 4 year tour of the saturnian system, the spacecraft executed a close flyby of the outer satellite Phoebe. The infrared channel of the Visual Infrared Mapping Spectrometer (VIMS) obtained images of reflected light over the 0.83-5.1 μm spectral range with an average spectral resolution of 16.5 nm, spatial resolution up to 2 km, and over a range of solar phase angles not observed before. These images have been analyzed to derive fundamental photometric parameters including the phase curve and phase integral, spectral geometric albedo, bolometric Bond albedo, and the single scattering albedo. Physical properties of the surface, including macroscopic roughness and the single particle phase function, have also been characterized. Maps of normal reflectance show the existence of two major albedo regimes in the infrared, with gradations between the two regimes and much terrain with substantially higher albedos. The phase integral of Phoebe is 0.29±0.03, with no significant wavelength dependence. The bolometric Bond albedo is 0.023±007. We find that the surface of Phoebe is rough, with a mean slope angle of 33°. The satellite's surface has a substantial forward scattering component, suggesting that its surface is dusty, perhaps from a history of outgassing. The spectrum of Phoebe is best matched by a composition including water ice, amorphous carbon, iron-bearing minerals, carbon dioxide, and Triton tholin. The characteristics of Phoebe suggest that it originated outside the saturnian system, perhaps in the Kuiper Belt, and was captured on its journey inward, as suggested by Johnson and Lunine (2005).     

Theoretical interpretation of the ground-based photometry of Saturn's B ring

New photographic photometry at small tilt angles during the 1979 and 1981 apparitions is combined with earlier data to yield several physical parameters for Saturn's B ring in red and blue colors. Phase curves are obtained for a mean tilt angle B ? 6°. The value of the volume density D is 0.020±0.004 with no indication of dependence on either the color or the tilt angle for 6°<B<26°. This conclusion is not altered significantly if the individual ring particles have a phase function similar to the phase curves of bright solar system objects. For the geometric albedo of a single particle we derive 0.61±0.04 (red) and 0.41±0.03 (blue), which are superior to earlier estimates because of the additional data now available. These values and the derived amount of multiple scattering as a function of tilt angle constrain the particle phase function in the red to be moderately backscattering. Inferred values of the particle single-scattering albedo are $\text{0.7≤}\text{ω}{}_0\text{(}\text{red}\text{) ≤0.92}$ and $\text{0.5≤}\text{ω}{}_0\text{(}\text{blue}\text{) ≤0.7}$, depending on the choice of phase function. No indication was found that the particle photometric properties might depend on the vertical distance from the central plane. Our results show that the ground-based photometry is entirely consistent with the classical, many-particle-thick ring model.     

Titan: Preliminary results on surface properties and photometry from VIMS observations of the early flybys

Cassini observations of the surface of Titan offer unprecedented views of its surface through atmospheric windows in the 1-5 μm region. Images obtained in windows for which the haze opacity is low can be used to derive quantitative photometric parameters such as albedo and albedo distribution, and physical properties such as roughness and particle characteristics. Images from the early Titan flybys, particularly T0, Ta, and T5 have been analyzed to create albedo maps in the 2.01 and 2.73 μm windows. We find the average normal reflectance at these two wavelengths to be 0.15±0.02 and 0.035±0.003, respectively. Titan's surface is bifurcated into two albedo regimes, particularly at 2.01 μm. Analysis of these two regimes to understand the physical character of the surface was accomplished with a macroscopic roughness model. We find that the two types of surface have substantially different roughness, with the low-albedo surface exhibiting mean slope angles of ∼18°, and the high-albedo terrain having a much more substantial roughness with a mean slope angle of ∼34°. A single-scattering phase function approximated by a one-term Henyey-Greenstein equation was also fit to each unit. Titan's surface is back-scattering (g∼0.3-0.4), and does not exhibit substantially different backscattering behavior between the two terrains. Our results suggest that two distinct geophysical domains exist on Titan: a bright region cut by deep drainage channels and a relatively smooth surface. The two terrains are covered by a film or a coating of particles perhaps precipitated from the satellite's haze layer and transported by eolian processes. Our results are preliminary: more accurate values for the surface albedo and physical parameters will be derived as more data is gathered by the Cassini spacecraft and as a more complete radiative transfer model is developed from both Cassini orbiter and Huygens Lander measurements.     

Venus: Cloud optical depth and surface albedo from Venera 8

We have used the measurements of the solar flux obtained by the Venera 8 spacecraft inside the atmosphere of Venus and the values of the Venus spherical albedo to deduce the characteristics of the clouds and of the ground. The method used is the exponential kernel approximation and the results have been tested by exact computations with the spherical harmonics method.A cloud layer with an optical thickness $\text{τ}{}_1\text{? 144}$, an albedo for single scattering $\text{ω}{}_0\text{= 0.9998}$ in the rear infrared, above a Rayleigh layer between 0 and 32 km and a ground of reflectivity ? = 0.4, gives a good agreement with the experimental results. A model with two cloud layers is also discussed.     

Phase function and polarization curve of interplanetary scatterers from zodiacal light photopolarimetry

Wide-angle ecliptic measurements of zodiacal light brightness (Z) and polarization (P) lead to fundamental results about optical properties of interplanetary scatterers, under a few reasonable assumptions (that they depend upon heliocentric distance by a r^?n law, and suffer no significant distortion of their scattering indicatrix between 0.5 and 2 a.u.): 1. The phase function σ(θ) is expressed (Equation 6) as a function of n and of (Z) data. 2. At the elongation ? = 90°, the derivative $\text{dZ}\text{d?}$ yields an absolute determination of the intensity $\text{T}$ scattered at right angles from the Sun by a single unit-volume of interplanetary medium (Equation 7). 3. The polarization degree $\text{P}$(θ) of the sunlight scattered by a single volume is derived (Equation 12) from n and from (Z + P) data. For two special values of the scattering angle θ, n vanishes in Equation (12), so that a fair knowledge of the polarization curve (Fig. 2) is reached prior to any assumption, or any forthcoming Jupiter-probe measure, about the value of n.Should n be provided by the Pioneers, then a thorough treatment of the whole problem of phase function and polarization curve can be performed by means of Equations (6) and (12) supplied with available zodiacal light photopolarimetric observations.     

Micropulsations and orthogonal curvilinear coordinates

The theory of torsional hydromagnetic oscillations of the magnetosphere is usually cast in terms of orthogonal curvilinear coordinates. For a general magnetic field B with potential Ω it is shown that no coordinates exist in which a suitable solution may be found unless the Alfvén velocity V_A, together with B and Ω satisfy certain functional relationships. In the case V_A = constant, for example we must have

$\text{(}\text{B}\text{· ?)B =}\text{function of}\text{B}\text{and}\text{Ω}\text{only}$

. The relationships presented are in fact satisfied by all the magnetic fields considered to date.     

Planetary spectra for anisotropic scattering

We examine here some of the effects on planetary spectra that would be produced by departures from isotropic scattering. The phase function $\text{}\text{?}\text{gw(1 + a}\text{cos}\text{θ)}$ is the simplest departure to handle analytically and the only phase function, other than the isotropic one, that can be incorporated into a Chandrasekhar first approximation. This approach has the advantage of illustrating effects resulting from anisotropies while retaining the simplicity that yields analytic solutions.The curve of growth is the sine qua non of planetary spectroscopy. Our discussion emphasizes the difficulties and importance of ascertaining curves of growth as functions of observing geometry. A plea is made to observers to analyze their empirical curves of growth, whenever it seems feasible, in terms of coefficients of $\text{(1?}\text{}\text{?}\text{gw)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and $\text{(1?}\text{}\text{?}\text{gw)}$, which are the leading terms in radiative-transfer analysis.An algebraic solution to the two sets of anisotropic H functions is developed in the appendix. It is readily adaptable to programmable desk calculators and gives emergent intensities accurate to 0.3%, which is sufficient even for spectroscopic analysis.     

Optical constants of organic tholins produced in a simulated Titanian atmosphere: From soft x-ray to microwave frequencies

As part of a continuing series of experiments on the production of dark reddish organic solids, called tholins, by irradiation of cosmically abundant reducing gases, the synthesis from a simulated Titanian atmosphere of a tholin with a visible reflection spectrum similar to that of the high altitude aerosols responsible for the albedo and reddish color of Titan has been reported Sagan and Khare, 1981, Sagan and Khare, 1982, Orig. Life. 12, 280) and [C. Sagan, B. N. Khare, and J. Lewis, in press. In Saturn (M. S. Matthews and T. Gehrels, Eds.), Univ. of Arizona Press, Tucson]. The determination of the real (n) and imaginary (k) parts of the complex refractive index of thin films of such tholin prepared by continuous D.C. discharge through a 0.9 N₂/0.1 CH₄ gas mixture at 0.2 mb are reported. For $\text{250}\text{A}\text{?}\text{≤ γ ≤ 1000 μ}\text{m}\text{, n}\text{and}\text{k}$ have been determined from a combination of transmittance, specular reflectance, interferometric, Brewster angle, and ellipsometric polarization measurements; experimental uncertainties in n are estimated to be ±0.5, and in k ± 30%. Values of n(?1.65) and k (?0.004 to 0.08) in the visible range are consistent with deductions made by ground-based and spacecraft observations of Titan. Maximum values of k (?0.8) are near 1000 Å, and minimum values (?4 × 10^?4) are near 1.5 μm. Many infrared absorption features are present in k(γ), including the 4.6-μm nitrile band.     

The albedo of Titan

The irradiance of Titan has been measured from 0.50 to 1.08μ in 30 Å band-passes spaced 0.01–0.02μ apart. Geometric albedos have been computed at the wavelenghts of measurement using a standard solar flux distribution after Labs and Neckel. The maximum value of p_λ(0) is 0.37 at 0.68, 0.75, and 0.834μ, the minimum value, in the centers of the strongest methane absorption bands, is 0.10 at 0.887 and 1.012μ.The brightness of Titan at the time of the present measurements has been compared with that of previous modern photoelectric measurements. Within the apparent consistency of the different photoelectric systems, the brightness of Titan appears to undergo changes with time.A provisional curve of the geometric albedo from 0.30 to 4.0μ has been made by combining the present results with those of other authors, i.e., relative measurements of Titan from 0.30 to 0.50μ, and measurements of Jupiter and Saturn from 1.08 to 4.00μ. The latter are used to estimate the strengths of the methane absorption bands of Titan in that spectral range. The bolometric geometric albedo, $\text{p}{}^1\text{(0)}$, is computed to be 0.21. A variety of current measurements of Titan indicate a substantial atmosphere, suggesting a value of the phase integral $\text{q}\text{= 1.30 ± 0.20}$. The bolometric Bond albedo, $\text{A}{}^1$, is then 0.27 ± 0.04, giving an effective radiative temperature T_e= 84 ± 2°K.The absorption band contours of Titan have been compared with those of Jupiter and Saturn at the same resolution. The bands of the planets are known to be due primarily to methane, and they show a very regular relationship, with those of Saturn being consistently deeper and wider. For Titan, the strengths of the bands are equal or less than those of Jupiter in the band centers, while the wings are stronger than those of Saturn.Previous photoelectric and photographic spectra have been examined for evidence of temporal variation of the methane path length in the atmosphere of Titan. Differences in measurement techniques prohibit detection of small differences. The only potential differences beyond experimental uncertainties are those of Kuiper (1944) and Harris (mid-fifties). Taking Kuiper's results at face value, Titan appears to have a shorter methane path length in 1972. Harris's results can be reconciled only by the doubtful hypothesis of an almost complete absence of methane at that time.     

Non-adiabatic processes in a two-fluid plasma and energization of the plasma sheet plasma

The change of energy of a collisionless, two-fluid plasma consists of the adiabatic gain or loss of energy, which is due to the work done by the electromagnetic forces, and of the non-adiabatic change associated with the presence of the “rest” field $\text{E}{}^1\text{=}\text{E}\text{+ (}\text{1}\text{c}\text{)}\text{V}\text{×}\text{B}$. The non-adiabatic gain or loss of energy per unit ti may be expressed by the relation

$\text{Q=}\text{E}{}_{\mathrm{\parallel }}\text{·}\text{i}{}_{\mathrm{\parallel }}\text{+}\text{c}\text{eNB}{}^2\text{B·}\text{f}\text{?}\text{×f}$

where i is the density of conductive current, N the ion number-density, and f (f?) the sum of inertia and pressure divergence of ions (electrons). Symbols of parallelism refer to the direction of B.A special case of non-adiabatic energization of a slowly convecting plasma sheet plasma is discussed in some detail. Regardless of the value of V, the non-adiabatic energization may significantly exceed any conceivable energization associated with the electric field $\text{?(}\text{1}\text{c}\text{)}\text{V}\text{×}\text{B}$.     

Effects of crossed magnetic and (spatially dependent) electric fields on charged particle motion

The motion of charged particles is examined in the case of a homogeneous magnetic field B together with an orthogonal electric field E, which has a gradient ▽E parallel to E. If

$\text{B}{}^2\text{q}{}^2\text{m}{}^2\text{?}\text{q▽E}\text{m}\text{> 0}$

, the particles drift at right angles to E and B with a modified gyrofrequency and produce a current in that direction. If

$\text{B}{}^2\text{q}{}^2\text{m}{}^2\text{?}\text{q▽E}\text{m}\text{< 0}$

, the particles not only drift in the direction of E × B but are also accelerated in the direction of E, in which direction they also produce a current.